1. science
2. mathematics
3. arithmetic

CSE 115 / ENGR 160
Spring 2015
Lecture 18
Announcements and Assignments
• Check course website for reading schedule
• HW #8 due Wed. Apr. 8 in class
• Midterm #2 on Wed. Apr. 15 during lecture
– More closer to time
Today
• 4.1 Divisibility and Modular Arithmetic (continued)
• 4.2 Integer Representations and Algorithms
Congruence Relation
Definition: If a and b are integers and m is a positive integer then a is congruent to b modulo m if m divides a – b.
 The notation a ≡ b (mod m) says that a is congruent to b
modulo m.  We say that a ≡ b (mod m) is a congruence and that m is its modulus.
 Two integers are congruent mod m if and only if they have the same remainder when divided by m.
 If a is not congruent to b modulo m, we write a ≢ b (mod m).
Example: Determine whether 17 is congruent to 5 modulo 6 and whether 24 and 14 are congruent modulo 6.
Solution: 

17 ≡ 5 (mod 6 because 6 divides 17 5 = 12.
24 ≢14 (mod 6 since 6 divides 24 14 = 10isnotdivisibleby6.
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More on Congruences
Theorem : Let m be a positive integer. The integers a
and b are congruent modulo m if and only if there is an integer k such that a = b + km.
Proof:  If a b (mod m) then (by the definition of congruence) m | a – b. Hence, there is an integer k such that a – b = km and equivalently a = b + km.
 Conversely, if there is an integer k such that a = b + km then km = a – b.
Hence, m | a – b and a b (mod m).
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The Relationship between (mod m) and mod m Notations
 The use of “mod” in a b (mod m) and a mod m = b are different.
 a b (mod m) is a relation on the set of integers.
 In a mod m = b, the notation mod denotes a function.
 The relationship between these notations is made clear in this theorem.
Theorem : Let a and b be integers, and let m be a positive integer. Then a b (mod m) if and only if a mod m = b mod m. (Proof in the exercises)
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Congruences of Sums and Products
Theorem : Let m be a positive integer. If a b (mod m) and c d (mod m) then
a + c b + d (mod m) and ac bd (mod m).
Proof:  Because a b (mod m) and c d (mod m), by Theorem there are integers s and t with b = a + sm
and d = c + tm.
 Therefore,  b + d = (a + sm) + (c + tm) = (a + c) + m(s + t) and
 bd = (a + sm) (c + tm) = ac + m(at + cs + stm).
 Hence, a + c b + d (mod m) and ac bd (mod m). 12
Congruences of Sums and Products
Example: Because (mod ) and it follows from Theorem that:
(mod ) ∙ ∙ (mod )
(mod ), 13
Algebraic Manipulation of Congruences
 Multiplying both sides of a valid congruence by an integer preserves its validity. If a ≡ b (mod m) holds then c∙a ≡ c∙b (mod m), where c is any integer, holds by Theorem 5 with d = c.
 Adding an integer to both sides of a valid congruence preserves its validity.
If a ≡ b (mod m) holds then c + a ≡ c + b (mod m), where c is any integer, holds by Theorem 5 with d = c.
 Dividing a congruence by an integer does not always produce a valid congruence.
Example: The congruence (mod ) holds. But dividing both sides by does not produce a valid congruence since 14
Computing the mod m Function of Products and Sums
 We use the following corollary to Theorem Corollary: Let m be a positive integer and let a and b be integers. Then
(a + b) (mod m) = ((a mod m) + (b mod m)) mod m
and
ab mod m = ((a mod m) (b mod m)) mod m. (proof in text)
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Arithmetic Modulo m
Definitions: Let Zm be the set of nonnegative integers less than m: { , , …., m }
 The operation +m is defined as a +m b = (a + b) mod m. This is addition modulo m.
 The operation m is defined as a m b = (a b) mod m. This is multiplication modulo m.
 Using these operations is said to be doing arithmetic modulo m.
Example: Find +11 and ∙11 .
Solution: Using the definitions above:
 7+11 9 7 9 mod 11 16mod 11 5
 7∙11 9 7∙ 9 mod 11 63mod 11 8
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Arithmetic Modulo m
 The operations +m and m satisfy many of the same properties as ordinary addition and multiplication.
 Closure: If a and b belong to Zm then a +m b and a m b
belong to Zm .
 Associativity: If a, b, and c belong to Zm then (a +m b) +m c = a +m (b +m c) and (a m b) m c = a m (b m c).
 Commutativity: If a and b belong to Zm then a +m b = b +m a and a m b = b m a.
 Identity elements: The elements and are identity elements for addition and multiplication modulo m, respectively.
 If a belongs to Zm , then a +m = a and a m = a.
continued →
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Arithmetic Modulo m
 Additive inverses: If a
belongs to Zm then m a is the additive inverse of a modulo m and is its own additive inverse.  a +m (m a ) = and +m =  Distributivity If a, b, and c belong to Zm then  a m (b +m c) = (a m b) +m (a m c) and (a m b) m c = (a m c) m (b m c).


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Section 4.2
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Section Summary
 Integer Representations
 Base b Expansions
 Binary Expansions
 Octal Expansions
 Hexadecimal Expansions
 Base Conversion Algorithm
 Algorithms for Integer Operations
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Representations of Integers
 In the modern world, we use decimal, or base
notation to mean represent integers. For example when we write 2
1
0.  We can represent numbers using any base b, where b is a positive integer greater than .
 The bases b = (binary), b = 8 (octal) , and b = (hexadecimal) are important for computing and communications
 The ancient Mayans used base and the ancient Babylonians used base .
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Base b Representations
 We can use positive integer b greater than 1 as a base, because of this theorem:
Theorem 1: Let b be a positive integer greater than 1. Then if n is a positive integer, it can be expressed uniquely in the form:
n = akbk + ak‐1bk‐1 + …. + a1b + a0
where k is a nonnegative integer, a0,a1,…. ak are nonnegative integers less than b, and ak≠ 0. The aj, j = 0,…,k are called the base‐b digits of the representation.
(We will prove this using mathematical induction in Section 5.1.)
 The representation of n given in Theorem 1 is called the base b expansion of n and is denoted by (akak‐1….a1a0)b.
 We usually omit the subscript 10 for base 10 expansions.
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Binary Expansions
Most computers represent integers and do arithmetic with binary (base ) expansions of integers. In these expansions, the only digits used are .
Example: What is the decimal expansion of the integer that )2 as its binary expansion?
has (
Solution:
8
7
6
5
4
3
(
)2
2
1
0
Example: What is the decimal expansion of the integer that )2 as its binary expansion?
has (
4
3
2
1
0
Solution: (
)2
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Octal Expansions
The octal expansion (base 8) uses the digits }.
{
Example: What is the decimal expansion of the )8 ?
number with octal expansion (
2
1
0
Solution: 3
Example What is the decimal expansion of the )8 ?
number with octal expansion (
1
0
Solution: 2
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Hexadecimal Expansions
The hexadecimal expansion needs digits, but our decimal system provides only . So letters are used for the additional symbols. The hexadecimal system uses the digits ,A,B,C,D,E,F}. The letters A through F {
represent the decimal numbers through .
Example: What is the decimal expansion of the number with )16 ?
hexadecimal expansion (
Solution: 4
3
2
1
0
Example: What is the decimal expansion of the number with hexadecimal expansion (E )16 ?
2
1
0
Solution:
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Base b Expansion
To construct the base b expansion of an integer n:
 Divide n by b to obtain a quotient and remainder.
n = bq0 + a0
a0 b
 The remainder, a0 is the rightmost digit in the base b
expansion of n. Next, divide q0 by b.
q0 = bq1 + a1
a1 b
 The remainder, a1, is the second digit from the right in the base b expansion of n.
 Continue by successively dividing the quotients by b, obtaining the additional base b digits as the remainder. The process terminates when the quotient is .
continued →
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Algorithm: Constructing Base b Expansions
procedure base b expansion(n, b: positive integers with b > 1)
q := n
k := 0
while (q ≠ 0)
ak := q mod b
q := q div b
k := k + 1
return(ak‐1 ,…, a1,a0){(ak‐1 … a1a0)b is base b expansion of n}
 q represents the quotient obtained by successive divisions by b, starting with q = n.
 The digits in the base b expansion are the remainders of the division given by q mod b.
 The algorithm terminates when q = is reached.
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Base b Expansion
Example: Find the octal expansion of (
)10
Solution: Successively dividing by 8 gives:

= 8 ∙ + 
= 8 ∙ + 
= 8 ∙ + 
= 8 ∙ + 
= 8 ∙ + The remainders are the digits from right to left yielding (
)8.
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Comparison of Hexadecimal, Octal, and Binary Representations
Initial 0s are not shown
Each octal digit corresponds to a block of 3 binary digits.
Each hexadecimal digit corresponds to a block of 4 binary digits. So, conversion between binary, octal, and hexadecimal is easy.
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Conversion Between Binary, Octal, and Hexadecimal Expansions
Example: Find the octal and hexadecimal expansions of (
)2.
Solution:  To convert to octal, we group the digits into blocks of )2, adding initial s as three (
needed. The blocks from left to right correspond to the )8.
digits , , , , and . Hence, the solution is (
 To convert to hexadecimal, we group the digits into )2, adding initial s blocks of four (
as needed. The blocks from left to right correspond to the . Hence, the solution is (
)16.
digits , , ,
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Binary Addition of Integers
 Algorithms for performing operations with integers using their binary expansions are important as computer chips work with binary numbers. Each digit is called a bit.
procedure add(a, b: positive integers) {the binary expansions of a and b
are (an‐1,an‐2,…,a0)2 and (bn‐1,bn‐2,…,b0)2, respectively}
c := 0
for j := 0to n 1
d := (aj + bj + c)/2
sj := aj + bj + c 2d
c := d
sn := c
return(s0,s1,…, sn){the binary expansion of the sum is (sn,sn‐1,…,s0)2}
 The number of additions of bits used by the algorithm to add two n‐bit integers is O(n).
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Binary Multiplication of Integers
 Algorithm for computing the product of two n bit integers.
procedure multiply(a, b: positive integers) {the binary expansions of a and b are (an‐1,an‐2,…,a0)2 and (bn‐1,bn‐2,…,b0)2, respectively}
for j := 0to n 1
if bj = 1then cj = a shifted j places
else cj := 0
co,c1,…, cn‐1 are the partial products}
p := 0
for j := 0to n 1
p := p + cj
return p {p is the value of ab}
 The number of additions of bits used by the algorithm to multiply two n‐bit integers is O(n2).
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Binary Modular Exponentiation
 In cryptography, it is important to be able to find bn mod m
efficiently, where b, n, and m are large integers.
 Use the binary expansion of n, n = (ak‐1,…,a1,ao)2 , to compute bn .
Note that:
 Therefore, to compute bn, we need only compute the values of b, b2, (b2)2 = b4, (b4)2 = b8 , …, and then multiply the terms in this list, where aj = 1.
Example: Compute 311 using this method.
Solution: Note that 11= (1011)2 so that 311 = 38 32 31 =
32 2 232 31 92 2∙9∙3 81 2∙9∙3 6561 ∙9∙3 117,147. continued →
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Binary Modular Exponentiation Algorithm
 This algorithm successively finds b mod m, b2 mod m, b4 mod m, …, mod m, and multiplies together the terms where aj = . procedure modular exponentiation(b: integer, n = (ak‐1ak‐2…a1a0)2 , m: positive integers)
x := power := b mod m
for i := to k
if ai= then x := (x power ) mod m
power := (power power ) mod m
return x {x equals bn mod m }
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